mfGeomPlanarShape: Planar Shape Space Geometry

In Almond-S/manifoldboost: Boosting Regression Models for Manifold Valued Data

Planar Shape Space Geometry

Description

Geometry of the 'classic' shape space identifying planar shapes with a centered and scaled complex vector on the complex sphere / complex projective space.

Super classes

manifoldboost::mfGeometry -> manifoldboost::mfGeomEuclidean -> manifoldboost::mfGeomUnitSphere -> mfGeomPlanarShape

Methods

Public methods

• mfGeomPlanarShape$new()

• mfGeomPlanarShape$structure()

• mfGeomPlanarShape$unstructure()

• mfGeomPlanarShape$unstructure_weights()

• mfGeomPlanarShape$structure_weights()

• mfGeomPlanarShape$register()

• mfGeomPlanarShape$register_v()

• mfGeomPlanarShape$log()

• mfGeomPlanarShape$transport()

• mfGeomPlanarShape$plot()

• mfGeomPlanarShape$validate()

• mfGeomPlanarShape$get_normal()

• mfGeomPlanarShape$clone()

Inherited methods

• manifoldboost::mfGeometry$distance()
• manifoldboost::mfGeomEuclidean$align()
• manifoldboost::mfGeomEuclidean$innerprod()
• manifoldboost::mfGeomUnitSphere$exp()

---

Method new()

Initialize planar shape geometry for data given in (long) FDboost format.

Usage

mfGeomPlanarShape$new(data, formula, weight_fun = NULL, arg_range = NULL)

Arguments

data

data in the format used in FDboost.

formula

formula describing the internal structure of the data, intended for the obj.formula of mfboost (interpreted by mfInterpret_objformula).

weight_fun

a function computing inner product weights for the geometry taking two arguments: arg, the argument of the functional data specified in formula, range the range of arg.

arg_range

the range supplied to the weight_fun.

---

Method structure()

convert 2D vectors (given in long format with dimension indicator) to complex

Usage

mfGeomPlanarShape$structure(y)

Arguments

y

a numeric vector representing an element of the manifold in an unstructured way. The position of elements and dimensions is obtained from the initialization of the geometry.

---

Method unstructure()

convert complex back to long format numeric

Usage

mfGeomPlanarShape$unstructure(y_)

Arguments

y_

a complex vector on the sphere, orthogonal to the constant.

---

Method unstructure_weights()

double inner product weights for complex vectors to appear in both dimensions in numeric long format

Usage

mfGeomPlanarShape$unstructure_weights(weights_)

Arguments

weights_

numeric vector of inner product weights matching the y_ as a complex vector.

---

Method structure_weights()

remove double weights to obtain inner product weights for complex vectors

Usage

mfGeomPlanarShape$structure_weights(weights)

Arguments

weights

numeric vector of inner product weights matching the y as a numeric vector. weights_ are duplicated for values of both dimensions.

---

Method register()

center y_ and scale it to unit norm.

Usage

mfGeomPlanarShape$register(y_)

Arguments

y_

a complex vector on the sphere, orthogonal to the constant.

---

Method register_v()

orthogonally project v_ into the tangent space of y0_.

Usage

mfGeomPlanarShape$register_v(v_, y0_ = self$pole_)

Arguments

v_

a complex tangent vector.

y0_

a complex vector on the sphere, orthogonal to the constant.

---

Method log()

Apply Log function of the sphere after rotation alignment

Usage

mfGeomPlanarShape$log(
  y_,
  y0_ = self$pole_,
  method = c("simple", "alternative")
)

Arguments

y_

a complex vector on the sphere, orthogonal to the constant.

y0_

a complex vector on the sphere, orthogonal to the constant.

method

alternatives "simple" and "alternative" for the expression used to compute the sphere Log-map. Passed to parent method.

---

Method transport()

for the parallel transport previous alignment is assumed, such that the transport of the sphere is directly applied.

Usage

mfGeomPlanarShape$transport(
  v0_,
  y0_,
  y1_,
  method = c("horizontal", "simple", "general")
)

Arguments

v0_

a complex tangent vector.

y0_

a complex vector on the sphere, orthogonal to the constant.

y1_

a complex vector on the sphere, orthogonal to the constant.

method

expression used for parallel transport: "general" corresponds to the one of Cornea et al. 2017, "horizontal" to the one of Dryden & Mardia 2012 p. 76, and "simple" to a simplified version of "general" where v0_ is horizontal to y0_. Passed to parent method.

---

Method plot()

default plotting function for planar shapes, plotting y_ in front of y0_ (after alignment).

Usage

mfGeomPlanarShape$plot(
  y_ = self$y_,
  y0_ = self$pole_,
  ylab = NA,
  xlab = NA,
  col = "black",
  ylim = range(Im(c(y_, y0_))),
  xlim = range(Re(c(y_, y0_))),
  pch = 19,
  asp = 1,
  yaxt = "n",
  xaxt = "n",
  type = "p",
  y0_par = list(col = "darkgrey", type = type),
  seg_par = list(col = "grey"),
  ...
)

Arguments

y_

a complex vector on the sphere, orthogonal to the constant.

y0_

a complex vector on the sphere, orthogonal to the constant.

ylab, xlab, xlim, ylim, xaxt, yaxt, asp

graphical parameters passed to base::plot with modified defaults.

col, pch, type

graphical parameters passed to base::plot referring to y_.

y0_par

graphical parameters for y0_.

seg_par

graphical parameters for line segments connecting y_ and y0_.

...

other arguments passed to base::plot.

---

Method validate()

check whether is.complex(y_).

Usage

mfGeomPlanarShape$validate(y_)

Arguments

y_

a complex vector on the sphere, orthogonal to the constant.

---

Method get_normal()

Obtain "design matrix" of tangent space normal vectors in unstructured long format.

Usage

mfGeomPlanarShape$get_normal(y0_ = self$pole_, weighted = FALSE)

Arguments

y0_

a complex vector on the sphere, orthogonal to the constant.

weighted

logical, should inner product weights be pre-multiplied to normal vectors?

---

Method clone()

The objects of this class are cloneable with this method.

Usage

mfGeomPlanarShape$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

Almond-S/manifoldboost documentation built on June 23, 2022, 11:06 a.m.